The standard formulation of quantum theory in finite dimensions is
an example of representing the state and effect spaces of a theory
as subsets of a vector space isomorphic to \(\mathbb {R}^{d+1}\) . Quantum states
are represented by density operators on \({d}\)  which form a convex
subset of the real vector space of Hermitian operators on \({d}\) ,
which is isomorphic to \(\mathbb {R}^{d^{2}}\) . Quantum effects can also be
embedded in this space with \(e\left(\omega \right)=\operatorname{Tr}\left(e\omega \right)\) 
for an operator \(e\)  satisfying \(0\le \langle \psi |e|\psi \rangle \rangle \le \langle \psi |\psi \rangle \) 
for all rays \(|\psi \rangle \in {d}\) . Using this representation of
the state and effect spaces is essential for \(d>2\)  since it is highly
non-trivial to explicitly describe the set of density operators by
vectors of the form given in Eq. (REF )) (see e.g.
[1]}, [2]}).
